Course Profile Foundations of Mathematics, Grade 10, Applied, Catholic

Unit 1: Modelling in Business and Finance

Time: 40 hours

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6 | Activity 7

Unit Description

In this unit, students will explore linear models in the context of applications in business and finance. Linear systems will be analysed both graphically and algebraically, with and without the use of technology. Activities in this unit address applications of piecewise linear functions; interpolation, extrapolation, finding and interpreting points of intersection, and solving linear systems by the methods of substitution and elimination. Students will develop proportional reasoning skills as they draw and interpret scale diagrams, and carry out investigations involving rate, ratio, and percent. Misleading graphical data and distortion of scale will also be examined. In preparation for Unit 2, students will explore a maximization problem that introduces the concept of quadratic functions and involves expanding binomial expressions.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE1d, 2b, 2c, 3c, 4a, 4f, 5a, 5b, 5g, 7b.

Strand(s): Proportional Reasoning and Linear Functions

Overall Expectations: PRV.01P, LFV.01P, LFV.02P, LFV.03P, QFV.03P.

Specific Expectations: PR1.01P, PR1.02P, PR1.03P, LF1.01P, LF1.02P, LF1.03P, LF2.01P, LF2.02P, LF2.03P, LF2.04P, LF3.01P, LF3.02P, LF3.03P, LF3.04P, QF1.01P, QF1.02P, QF3.02P.

Activity Titles (Time + Sequence)

The activities in this unit are designed to allow students to make connections between the mathematics classroom and the practical world of business and finance. The timelines provided are suggestions to guide teacher planning and can be modified to suit the needs of individual classrooms. Teachers may consider including locally relevant applications of the topics covered in these activities, where appropriate. The activities provided involve considerable group work, and thus provide the student with opportunities to demonstrate Catholic values as collaborative contributors and Christian leaders. Many of the activities also provide opportunities for students to demonstrate their understanding of responsible citizenship and societal awareness.

Activity 1.1 Follow-up Skills	Cyber Blueprints [scale drawings; percent change; conversions (currency and measurement); area; proportional reasoning; distortion] Interest calculations; estimation	375 minutes 75 minutes
Activity 1.2 Follow-up Skills	Getting Paid [graphical interpretation; interpolation, extrapolation; slope as a rate; graphing from tables of values] Descriptive work with linear and piecewise linear graphs (e.g., gas bill analysis; distance-time graphs); misleading graphical data	90 minutes 285 minutes
Activity 1.3 Follow-up Skills	Peace and Development Fundraiser: Intersection of Linear Models [construction of linear graphs from tables of values; interpolation; intersection of lines; formula development and evaluation (word and algebraic equations)] Solving 1st-degree equations; equations with fractional coefficients; working with formulae	90 minutes 285 minutes
Activity 1.4 Follow-up Skills	Bottled Water Dilemma [use of technology in graphing; intersection of lines; subtraction of equations] Finding the intersection of lines using graphing calculators or graphing software; solving contextual problems	75 minutes 150 minutes
Activity 1.5 Follow-up Skills	The Rewards of Design [algebraic method of elimination] Practise with the elimination method; introduce method of substitution; re-arrange equations from the form y = mx + b to the form Ax + By + C = 0 and vice versa]	150 minutes 300 minutes
Activity 1.6 Follow-up Skills	Battle of the Bands – Introduction to Quadratics [construction of graphs from tables of data; maximization of revenue and area; graphing a product of binomials; quadratic regression; connect product of binomials with ax^2 + bx + c] Expand and simplify polynomial expressions involving the multiplying and squaring of binomials	150 minutes 150 minutes
Activity 1.7	Summative Activity	225 minutes

Prior Knowledge Required

Number Sense and Algebra

· manipulate 1st-degree polynomial expressions to solve 1st-degree equations (excluding equations with fractional coefficients)

· add and subtract polynomials; multiply a polynomial by a monomial; expand and simplify polynomial expressions involving one variable

· solve problems, using the strategy of algebraic modelling

· solve problems involving applications of percent, ratio, and rate

Relationships

· determine relationships between two variables by collecting and analysing data

· compare the graphs of linear and non-linear relations

· collect, organize, and analyse data using appropriate equipment and/or technology

· describe trends and relationships in data

· construct tables of values, graphs, and formulas from descriptions of realistic situations, and from data collected experimentally

· use interpolation and extrapolation to gather information from a graph

· distinguish between linear and non-linear relations by calculating finite differences

Analytic Geometry

· identify the properties of line segments (direction, positive/negative slope, parallelism, perpendicularity)

· calculate slope using formula m = rise/run

· graph lines by hand and using graphing technology

· determine the equation of a line given slope and y-intercept, slope and a point on the line, and two points on a line, in the form y = mx +b

Unit Planning Notes

· The first activity is a multi-step analysis of properties of scale, rate, ratio, and percent. It is recommended that assessment and evaluation be carried out in appropriate stages to reduce marking volume.

· The use of computer-aided design programs may be considered in Activity 1. Students should be given appropriate instruction if such programs are to be used. Appropriate scheduling of computer labs may be necessary.

· The activity Getting Paid provides an opportunity for students to carry out independent research into the payment schedules of various careers. This research time may be assigned for homework, or may be incorporated into a resource period. If resource time is provided, the appropriate facilities should be scheduled in advance.

· Practise using spreadsheets and graphing calculators in the context of the activities presented.

· Prepare to diagnose prior learning skills throughout the unit. Some students may require remediation in order to meet standard expectations. Skill development activities should be developed to meet the needs that arise.

· Some students may require extended or enrichment activities to challenge their learning.

· The activities in this unit provide opportunities to integrate other disciplines and to incorporate locally relevant examples. Every effort should be made to link the topics presented in this unit with the practical world beyond the classroom.

Teaching/Learning Strategies

· It is expected that direct, teacher-lead instruction will be integrated within the framework of the activities as required to facilitate student learning and success. Independent practice of new skills will be necessary throughout the course.

· It is recommended that students be assigned to groups with any special needs and strengths considered. Appropriate peer grouping to benefit those students requiring extra help is suggested. If a peer tutor program is available at your school, tutors should be matched with those students requiring extra attention.

· Students will be involved in a considerable amount of group work. It would be beneficial to take some time to review appropriate group work dynamics, sharing of work responsibilities, assigning group roles, etc.

· Students should be encouraged to take ownership and responsibility for their own learning.

· Appropriate opportunities for students to communicate solutions, ideas, and concepts should be provided throughout the course.

Assessment and Evaluation

· It is recommended that students be involved in the development of some of the rubrics used in the assessment/evaluation process. Students should be encouraged to self-evaluate and to identify areas that need improvement.

· Sample generic rubrics for oral presentations, written reports, and various learning skills are provided in this profile. They may be adapted to suit the needs of your classroom.

· Rubrics should be used as often as possible to assess student work. They are particularly appropriate to assess the expectations under the thinking/inquiry/problem-solving, communication, and application sections of the achievement chart.

· When rubrics are used in assessment, students should be provided with the specific rubric that will be used prior to completing the assigned task.

· Care should be taken in the design of traditional paper and pencil tests to ensure that a level 4 performance can be demonstrated.

· This unit incorporates both a final assessment activity and a paper and pencil test. Recommended evaluation rubrics are also provided.

· Work in partnership with the special education department to develop strategies to meet the IEP recommendations

· Accommodations allow students with special learning needs to meet the expectations of the course

· Learning skills should be assessed in conjunction with the academic skills for each activity.

Resources

Internet

Computer-aided design software (optional)

Major newspaper (business section for currency tables)

Graphing calculators; graphing software

House plan/design books

NCTM Addenda Series: Example 3 and Activity 2 from “Algebra in a Technological World”

The Mathematics Teacher, Vol. 88 #3, March 1995: “Gas Bill Mathematics”

Lappan, Glenda, et al. Moving Straight Ahead Linear Relationships. Dale Seymour Publishing, 1988.

NCTM: Activities for Active Learning and Teaching.

Balanced Assessment for the Mathematics Curriculum. Harvard Press.

Choices into Action Ministry Curriculum Document

Activity 1.1: Cyber Blueprints

Time: 375 minutes

Description

This introductory activity addresses the areas of scale; rate; ratio; area; percent, measurement, and currency conversions in the practical context of a web-based blueprint design business. Opportunities are provided in this activity to incorporate such technology as computer-aided design programs and presentation software. Students work in partners to prepare a written report to address the questions raised in the activity and deliver an oral presentation in the form of a business report to highlight their findings.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

- a reflective and creative thinker who evaluates situations and solves problems;

- a self-directed, responsible, life-long learner who applies effective communication, decision-making, problem-solving, time, and resource management skills;

- a collaborative contributor who works effectively as an interdependent team member achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others.

Overall Expectations

PRV.01P - solve problems derived from a variety of applications using proportional reasoning.

Specific Expectations

PR1.01P - solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, and direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations);

PR1.02P - draw and interpret scale diagrams related to applications;

PR1.03P - distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning).

Planning Notes

· This is a week-long project involving multiple activities related to the design of a home and the following time frame is suggested:

Part A: The Design Process (110 minutes)

Parts B and C: Currency Conversion/Re-scaling the Design (115 minutes)

Part D: Presentation of Report (75 minutes)

Part E: Distortion (75 minutes)

· Reserve computer time to provide access to design software and Internet.

· Students should work in pairs or small groups – the classroom should be organized accordingly.

· Provide up-to-date currency conversion charts from the business section of a major newspaper.

· Share sample house plans with the class prior to the activity (see Appendix).

· If computer-aided design programs are to be included, a mini-workshop to facilitate student use is recommended prior to this activity (see Technology Departments for information).

· Scientific calculators are required.

· Students should keep daily logs (journals) of their progress – this can be collected with the written report.

· To facilitate marking, student work should be collected in stages.

· This activity has cross-curricular connections with computer science, business, and technological (i.e., drafting) courses – consultation with these departments may be beneficial

Prior Knowledge Required

Number Sense and Algebra

· determine strategies for mental mathematics and estimation and apply these strategies

· demonstrate facility in operations with percent, ratio and rate

· use of a scientific calculator and graphing technology effectively

· judge the reasonableness of solutions

Measurement and Geometry

· solve problems involving area

Teaching/Learning Strategies

Cyber Blueprints

Teacher Facilitation: Pose the following scenario to the class:

You are a young entrepreneur who has developed a web-based business to market and sell original house plans to a world-wide customer base. In addition to designing a variety of standard house plans, you also accept requests for custom-designs from your clients.

You have a recent request from a customer in England to design a house to meet the following needs:

1. The foundation of the house must occupy 50% of the lot, which measures 60' wide by 120' long; a minimum of 5' must be left on either side of the house to allow passage to the back yard.

2. The house must be a bungalow model with adequate space to accommodate the family with three young children.

3. The husband is an avid cook and spends considerable time in the kitchen.

4. The wife runs a home-based business and requires space to set up a home office.

Student Activity

Students investigate scale diagrams, rate, ratio, percent; area and conversions in measurement, and currency as they explore a web-based house design business.

Part A: The Design Process

1. Design a suitable house plan to meet the above needs and draw the plan, to scale, to reflect your design. Discuss the factors you considered in developing this plan.

2. Calculate the total area of living space in the home in square metres. What difficulties did you face in calculating the area of this home? How did you overcome these difficulties?

3. Determine what you will charge your client for the house plan (in Canadian dollars). What factors did you consider in setting your price? Was there a formula you used?

Teacher Facilitation: Lead class through sample scale diagram constructions and percentage calculations. Brainstorm factors to consider when planning home designs. Sample home designs may be shared and analysed.

If designs are to be drawn using computer-aided design programs (such as CAD or CorelDRAW™), ensure that students have been given the appropriate instruction on the use of the program. You may need to liaise with a Technological Education teacher at your school.

Where such technology is unavailable, students may construct scale diagrams on graph paper.

Students may require practise in the calculation of area for regular and irregular shapes.

During the design process, conference with students to assess progress and provide guidance to those having difficulty with scale drawings.

Extension: students requiring an additional challenge may consider designing a two-story house plan instead of a bungalow.

Part B: Currency Conversion

1. Your client is pleased with your design, and has decided to purchase your plan. However, he requests that you convert the price to British pounds. Show your calculations.

Alternate Activity: Place students in groups and assign each group a particular country to consider for the currency conversions. Have each group report their findings to the rest of the class, with an explanation of their calculations. A whole-class comparison of the currency conversion differences between countries could be examined and analysed.

Teacher Facilitation: Provide a mini-lesson on currency conversion using tables provided from the newspaper or Internet (this may be done in context of the activity or using the scenario of travel to highlight applicability of knowledge of currency conversion).

Part C: Re-scaling the Design/Conversions

You are quite happy with the design you created and decide to post it on your web site to see if other customers might be interested in purchasing it.

A client in the United States has expressed interest in the plan but has the following requests:

1. Re-scale the plan so that the dimensions of the home are increased by 15%. Draw a new scale diagram to illustrate this change (use computer-aided design program if possible).

2. Report the new area in both square metres and square feet (as the US uses the Imperial system of measurement).

3. Report the cost of the drawings in US dollars (show calculations used).

4. Will you consider offering volume pricing for customers, such as contractors, who may be interested in purchasing multiple plans? If so, how will you determine volume pricing? If not, explain why not.

Teacher Facilitation: Provide a mini-lesson on percent change in the context of measurement applications and discuss the concept of volume pricing with appropriate examples.

Part D: Presentation of Report

1. Prepare a written business report to summarize your work on this project. The report will include your scale diagrams, pricing options, customer requests, and conversion calculations.

2. Prepare an oral report for the class. You may consider the use of presentation software to enhance your report. If presentation software is unavailable, you will display your drawings in an appropriate manner. Your report should be a creative presentation of your work - you may consider developing your presentation around a business meeting format where you will “sell” your plans to potential customers in the classroom.

Part E: Distortion as a Result of Irregular Scale

Teacher Facilitation: Provide students with examples of drawings with misleading scales. Discuss relationship between scales and misrepresentation of information. An example is provided.

Scale Worksheet

The following scale diagrams illustrate sketches of a proposed deck for the house plans you have designed in this activity.

1. (a) View the scale diagrams shown above. Which deck do you think has the largest actual area? Explain your answer.

(b) Calculate the area of each figure. Do your answers support your explanation given in A? Why or why not?

2. Draw your own scale diagram to illustrate how scale drawings can be misleading to the viewer. Construct questions based on your scale diagrams that you could ask a classmate to answer.

Teacher Facilitation: Extend discussion to area of home design: the scenario when house plans and actual home does not match due to irregularity in scale of design. This may lead to a discussion of consumer manipulation by advertisers due to distorted data.

Follow-up Skills: 75 minutes

· Further practise with percentage calculations in areas such as interest/investment calculations.

· Encourage development of estimation skills as they apply to proportional reasoning.

· Practise recognizing various types of distortion and misleading information in graphs.

Assessment/Evaluation Techniques

· Assess Learning Skills areas of team work, initiative and organization as students work in partners or small groups on stages of project (see rubrics provided in Appendix A).

· Collect and evaluate written report using written report rubric (emphasis on Thinking/Inquiry and Application)

· Evaluate Communication skills using oral presentation rubric.

· Peer Assessments using checklists; rubrics (may be used for group work and/or oral presentations) Sample rubrics are provided in the Appendices.

· Paper and Pencil Tasks to assess Knowledge and Understanding (e.g., quizzes on scale drawing, percent change, conversions, distortion, etc.)

Accommodations

· Groupings should be heterogeneous (pair students having difficulties with students who can provide guidance).

· Utilize peer tutors to assist students with special needs.

· Allow for alternative submissions of work where appropriate (e.g., video presentations to replace oral report in front of class; scribing of written reports, etc.)

Extensions

· Have students design a web page to advertise their business.

· Work in conjunction with the technology department (drafting) if students are interested in developing complex house designs.

Resources

House design books (see local building centre)

Business section of major newspaper (currency information)

Internet sites pertaining to home design (general search using any search engine)

Generic rubrics for evaluation of group work, written reports, oral presentations

Activity 1.2: Getting Paid

Time: 90 minutes

Description

In this activity students will construct tables of values and graphs of piecewise linear functions which represent earnings over time for people in various occupations who are paid in a variety of ways. Working in groups, students will use graphs of piecewise linear functions to describe the variation in earnings, over a short term and a long term, for individual case studies. Within the case studies, students will observe the effects of changes in earnings over time on the graphs. Students will also discuss the implications of this variation for short term budgeting and long term financial planning. Group presentations to the class will allow for comparison of graphs for different payment methods. Note that this activity has a career awareness component.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands, and uses written materials effectively;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Linear Functions

Overall Expectations

LFV.01P - apply the properties of piecewise linear functions as they occur in realistic situations.

Specific Expectations

LF1.01P - explain the characteristics of situations involving piecewise linear functions (e.g., pay scale variations, gas consumption costs, water consumption costs, differentiated pricing, motion);

LF1.02P - construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software;

LF1.03P - answer questions about piecewise linear functions by interpolation and extrapolation, and by considering variations on given conditions.

Planning Notes

· These activities may be completed using graphing calculators, spreadsheets, or paper and pencil. You must decide ahead of time which medium to use and plan to have the necessary equipment (graphing calculators and projector, computer lab and projection device, graph paper, chart paper, and markers) available for the class.

· Although a warm up activity is included, you may wish to precede this activity with a lesson spent on matching scenarios to given graphs and creating graphs for given scenarios. These should include, but not be restricted to, piecewise linear graphs.

· Connect with the Career Education and Guidance department in your school and obtain any flyers, posters, or other reading materials that provide information about the jobs you will be featuring in the main activity. Consult the ministry document Choices into Action for further connections.

· To make the activity more realistic for the students in your class, include some local occupations. You may need to do some research ahead of time to find out how the people in these occupations are paid.

· You will need to provide information on the methods of remuneration for a variety of occupations in the form of case studies (see Appendix A or you may wish to create your own or have students research the required information).

Prior Knowledge Required

Relationships Grade 9 Applied

· organize and analyse data using appropriate techniques and technology; describe trends and relationships observed in data, make inferences from data, compare the inferences with hypotheses about the data, and explain the differences between the inferences and the hypotheses; communicate the findings of an experiment clearly and concisely, using appropriate mathematical forms (e.g., written explanations, formulas, charts, tables, and graphs) and justify the conclusions reached.

Analytic Geometry Grade 9 Applied

· graph lines by hand using a variety of techniques; graph lines using graphing calculators or graphing software.

Teaching/Learning Strategies

Student Activity

Have students work on a 15-minute “warm up” activity (in which they will use some of the knowledge and skills gained in Grade 9 Mathematics) that involves the interpretation of a piecewise linear graph depicting the relationship between earnings and hours worked for someone who is paid at a fixed hourly rate for the first forty hours and then at time-and-a-half for hours over forty.

Ask students the following questions:

· How are Maria’s earnings related to the number of hours she works?

· How much does Maria earn if she works for 25 hours? for 65 hours?

· What is her hourly rate?

Then have students do the following:

· Mike is paid $9.50/h for working a 35-hour week, and time-and-a-half for any hours he works over 35 hours. Make a rough sketch of what you expect the graph of Mike’s weekly earnings would look like.

· Create a table of values of Mike’s earnings with hours worked up to sixty hours.

· Draw a graph of this data on the same grid as Maria’s earnings.

Teacher Facilitation: Important points to note, in taking up this warm up, are

· earnings increase as the number of hours worked increase;

· the slope of each linear part of the graph gives the hourly rate during that time period;

· when the hourly rate increases, the slope increases.

If you are using a graphing calculator or spreadsheet you could have the graph projected and ask students to create a table of values, on a calculator or computer spreadsheet, to match the graph you are projecting. They would then proceed to answer the questions above.

Whole class discussion: As an introduction to the main activity, spend 5-10 minutes brainstorming different ways people are or might be paid for the work that they do. In the discussion make connections between types of jobs and method of payment. Ask students for their ideas on why certain people are paid in a certain way, e.g., Why does commission form a large part of a salesperson’s earnings?

Student Activity

Working in groups of three or four, students will perform a numerical and graphical analysis of earnings over two different periods of time, in the case of a particular occupation, based on information provided by the teacher. Each group will consider a different occupation.

Student Worksheet: Getting Paid

How people are paid for the work that they do

1. Read the career case study you have been given

< Insert a sample Case Study (from those listed at the end of the Activity 1.2) here >

2. Create a table of values and a graph to depict earnings for one year for your given case.

(i) What are the variables?

(ii) Which is the independent variable? which is the dependent variable?

(iii) What scale should be used on the horizontal axis? the vertical axis?

(iv) Does the year being depicted begin in January or in some other month?

3. Describe and explain any trends, relationships, and special features in the graph you have created.

4. How do you think this method of payment would affect the way in which this person plans his/her weekly and/or monthly spending

5. Invent a career path (e.g., maternity leaves, layoffs, promotions, etc.), for the next ten years, for the person in your case study. Create a table of values and a graph depicting annual income over a period of ten years. You may be as creative as you like so long as your graph matches your invented career path.

6. How does the information on this graph affect long term financial planning for this individual?

7. Prepare a five-minute presentation of your case to the class, in which your group will:

· show the graphs you created;

· discuss the graphs and the group’s responses to Questions 3-6;

· discuss advantages and disadvantages, from a budgeting point of view, of the method of payment depicted.

Teacher Facilitation: As students work on the activity, the teacher should circulate and provide help where needed. Students will likely need reassurance and/or assistance with their responses to Questions 3, 4, 5, and 6 before they are ready to present to the class. During the presentations the teacher prompts the presenters with probing questions to ensure that all features of the graphs are fully explained. Students are also encouraged to pose their own questions to the presenters. (As an alternative to presentations, conduct a whole-class discussion, with the graphs posted at the front of the classroom for easy analysis.)

If students have been working with graphing calculators or computer spreadsheets, have them present their graphs using the technology. You may then ask them for hard copies to post in the classroom

Follow-up Skills: 285 minutes

· Students should now be given the opportunity to do more descriptive work with linear and piecewise linear graphs obtained when fuel consumption costs are considered (see Resources). Some review and extension of distance-time graphs would also fit in here.

· See the Harvard assessment task booklets for more practice with piecewise linear functions e.g., “Ford and Ferrari” and “Dollar Line” both from their Grade 10 booklet.

Extension: Consult the Stats Canada web site for information on average Canadian earnings by gender and education level; discuss implications with class.

Assessment/Evaluation Techniques

· Assessment in the Learning Skills areas of Teamwork, Independence and Initiative is possible as students work on the main activity.

· Individual student communication skills may be assessed during the presentations.

· If students are compiling a career portfolio, they could be asked to do some independent research into the career they have considered in this activity and to write a report which includes their group’s earnings analysis. This could then be assessed in the categories of knowledge/understanding and communication.

Resources

Example 3 and Activity 2 from “Algebra in a Technological World,” NCTM Addenda series

“Gas Bills Mathematics.” The Mathematics Teacher, Vol. 88 # 3, March 1995.

Choices into Action Ministry of Education Curriculum Document

Harvard Balanced Assessment Project.

Appendices

Rubric for Learning Skills Observation

Rubric for Presentations

Case Studies: “Getting Paid”

Teacher Facilitation: The following are examples of possible career case studies that could be used in this activity. It is recommended that individual teachers supplement these scenarios with locally relevant examples. These case studies provide the basis for the graphs that students will construct in the activity. Teachers may attach the scenarios to the worksheet provided to guide student work or the activity may be left more open-ended, allowing for student research and questioning.

Marion is a car salesperson. She earns a base salary of $400 and receives 3% of the value of each car she sells as commission. She must sell a minimum of two cars a week to keep her job. April, August, and December are busy months at the car dealership where she works – they sell 30% more cars in these months than at other times. The average selling price for a car from this dealership is $22 000. Marion is married and has a son and a daughter, both in their teens. Her husband is disabled and does not work outside the home.

Sue works part-time at a donut store and is paid $8.50/h for working weekdays and $10/h for weekend work. She is a university student and must save to pay her tuition fees which are due in September and in January. During the school year she works a six-hour shift two evenings in the week and eight hours on Sunday. From May to August she works a 40-hour week and also some weekends when the store is really busy. Sue has two more years to go before she earns her degree. After that she hopes to secure an intern position as a physiotherapist at the local hospital. Her starting salary would be $32 000 rising by annual increments of $3000 to a maximum of $65 000.

Mary works at the Wiblets corn processing plant. She is paid by piecework, receiving one cent for every cob of corn she processes. On average, she can process ten cobs of corn per minute, and works 12 hours per day in shifts of three days on, three days off. She works this schedule for the peak corn season during August and September. For the remainder of the year, she is a casual labourer at the plant, working an average of 20 hours per week at minimum wage. During this low season, Mary takes night courses at the local community college. She hopes to get her diploma in early childhood education and wants to open her own daycare in the future.

Doug works for a landscape firm from May until November. The firm only sends people out on non-rainy days and Doug is paid $85 for each day that he works. He is laid off at the end of November and re-employed the following May. In the layoff period he runs a one-man snow removal company and has ten small businesses as clients. He charges a flat rate of $200 for clearing a parking lot. Doug is married and has an 18-month-old son.

Mike serves tables at a mid-price restaurant from noon until 8:00 pm on weekdays and from 1:00 pm to 9:00 pm on Saturdays. He is paid minimum wage but receives tips. The average lunch costs $12.00 and the average dinner is $16.00. He can expect tips in the range of 10-25% of the total bill. The restaurant, which is close to the stadium, is busy during the baseball season and is packed in the weeks leading up to Christmas. The rest of the year business tends to be fairly slow. Mike lives at home with his mother, who is elderly and infirm.

Kiranjit is a supply teacher. She is paid a flat rate of $130/day. During the ‘flu season’ from mid-October to Christmas, she is able to work five days a week. At other times in the school year, she works two or three days a week. In July, she teaches at summer school where she works five hours a day for 20 days and is paid $35/h. Kiranjit is single and shares a loft apartment with three friends.

Mary Rose owns and operates a computer repair business. She charges $60 per hour for her labour. Business is fairly steady, she usually works for 10-12 hours each day except for July and August which are quieter. In these months Mary Rose only repairs 6-8 computers per week spending an average of 1.5 hours per computer. She makes up for this in September when she often has to work for 14 hours in a day.

Said is a bricklayer. The amount he earns depends on the number of bricks he lays. He is paid $750 for 1000 bricks and he usually lays 450 bricks in a day. He is really busy from May to October, working seven days a week, but for the rest of the year he only averages two to three days work per week. Said supports his wife and children who are still in Iran. He sends money to them on a monthly basis.

Frank is an accountant with a large investment company. His salary is $63 000/a and he receives a bonus when the company exceeds its target for the quarter. If paid, these bonuses, which range in amount from $500 to $5000 occur in March, June, September and December. He makes a point of donating 20% of each bonus to his favourite charity. Frank has a wife and three children and also supports his two children by a previous marriage.

Kasia is a newly-hired police officer in rural Ontario. Her salary, which is $35 000/a now, will rise in increments of $2700 for the next eight years. She is expecting a baby in five months and plans to take 18 months off when her baby is born. She will be paid for the first six months of this leave by employment insurance and the remainder of her leave will be without pay.

Activity 1.3: Peace and Development Fund Raiser: Intersection of Linear Models

Time: 90 minutes

Description

By developing tables of values, sketching graphs, writing linear equations, interpreting the point of intersection students will analyse a situation with variations. In this activity, the student council of Holy Mary High School has decided to do a walk-a-thon as one of its many fund raising activities and are faced with a decision of an appropriate, consistent donation plan for the whole student body to follow.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- a discerning believer formed in the Catholic faith community who develops attitudes and values founded on Catholic social teaching and acts to promote social responsibility, human solidarity, and the common good;

- a self-directed, responsible learner who demonstrates a confident and positive sense of self and the respect for the dignity and welfare of others;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Linear Function and Proportional Reasoning

Overall Expectations

LFV.02P - solve and interpret systems of two linear equations as they occur in applications;

LFV.03P - manipulate algebraic expressions as they relate to linear functions.

Specific Expectations

LF1.02P - construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software;

LF1.03P - answer questions about piecewise linear functions by interpolation and extrapolation, and by considering variation on given conditions;

LF2.01P - determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P - interpret the point of intersection of two linear relations within the context of a realistic situation;

LF3.01P - write linear equations by generalizing from tables of values and by translating written descriptions;

PR1.01P - solve problems involving rate and percent (e.g., in topics such as direct and partial variation related to linear functions) by a variety of methods and models (e.g., tables, patterns, graphs, equations).

Planning Notes

· The teacher will obtain a class set of graphing calculators and graph paper.

· The teacher will discuss any particular charity their school usually supports and the responsibility of all citizens to reach out to others in need. The chaplaincy office or the religion department in your school would be a place to obtain more information on this topic.

· There could be opportunity to extend this activity later in this unit by using the other fund raising suggestions given by the students and solving systems of linear equations by algebraic methods.

Prior Knowledge Required

Relationships

· use a graphing calculator or graphing software on a computer

· construct tables of values and graph linear relations

· construct formulas derived from descriptions of realistic situations involving direct and partial variation

· determine the values of a linear relation by formula and by interpolating or extrapolating from the graph

Analytic Geometry

· describe the effect on the graph and the formula of a relation by varying the conditions of the situation they represent

Teaching/Learning Strategies

Teacher Facilitation: The students will work in small groups of two or three. The teacher will help with any group experiencing difficulties in Part A of the activity or stop the class in order to allow students to share their comments and ideas about the task at hand. Students should be able to complete Part B with very little help from the teacher. During Part B, the teacher will be able to observe and encourage students to work efficiently within a group setting. The students should be prepared to hand in this report and/or to give a quick oral presentation using chart paper or overhead sheets to illustrate their charts, graphs and equations.

Student Activity

The students will work in small groups of two or three to complete the following handout.

The Peace and Development Fund Raiser – The Student Handout

The student council of Holy Mary High School decided to encourage the students to participate in a ten kilometre walk-a-thon as one of the many fund raising activities for the Peace and Development Fund. At a meeting to decide on a fair donation rate per kilometre to ask of the sponsors, Andrew, the student council president, stated that the students in his home room suggested 75 cents per kilometre. Beth, who is the representative of another class, stated that perhaps the students can ask for a $5.00 donation plus 25 cents per kilometre. All agreed that each walker must have a minimum of five sponsors on their pledge sheet.

Part A

1. Make a table showing the amount of money which would be pledged under each plan if the students walk up to 10 kilometres.

2. Using different colours, graph each pledge plan on the same coordinate axes.

3. At what point do the two lines intersect each other? Explain what this means in the context of this situation.

4. For each plan, write a formula that will help the volunteer calculate the amount of money one sponsor owes, given the distance the student completed. Write the formula in words first and then in algebraic form.

5. If the student completed 7.5 kilometres, how much would the sponsor owe under each plan.

a) Explain how you got the amount owed using the graph.

b) Show your calculations using the formula of Question 4.

c) Are the amounts in part a) and b) the same or different? Why?

6. a) If the sponsor owes $6.00, how many kilometres would the student have walked under each plan? Explain how you found the distance.

b) If the sponsor owes $7.20, how many kilometres would the student have walked under each plan? Explain how you found the distance.

7. Beth suggested a $5.00 donation and then 25 cents per kilometre.

a) How is this $5.00 donation represented on the graph?

b) How is the rate of 25 cents per kilometre represented on the graph?

c) If the rate of 25 cents per kilometre changed to 50 cents per kilometre, how would this change the graph? Using a different colour draw the line on the same axes as Question 2. Don’t forget to show a table of values for this situation and state the formula.

d) The line in part c) intersects the line representing Andrew’s plan. What is the point of intersection and explain what it means.

8. By changing the initial donation or the rate of donation or both, find a new pledge plan which will give a total of $18 for 7 kilometres walked. State the formula in algebraic form. Graph this new pledge plan on a separate axes from Question 2.

9. Which pledge plan would you suggest Holy Mary High School use in this walk-a-thon? Give reasons for your choice.

Part B

1. Your home room does not like the plans suggested so far.

a) Describe two other plans which could be used to raise funds in the walk-a-thon. The only restriction student council has placed on the pledge forms is that no sponsor should pay more than $20.00 for the completed distance.

b) Make tables of values and graphs to illustrate these two plans.

c) Write a formula for each.

d) Do the two lines intersect each other? Where? Explain what the intersection point means.

e) Pick a distance before the intersection point and describe which plan is better. Explain why.

f) Pick a distance after the intersection point and describe which plan is better. Explain why.

g) From these two plans, which one would you prefer and why?

2. Using your chosen plan, what would the minimum amount of pledges be, if 20% of 1500 students from Holy Mary High School finished the walk-a-thon.

3. What other activities could this school use to raise money for the Peace and Development Fund.

Follow-up Skills: 285 minutes

Specific Expectations

· LF3.03P- solve first degree equations in one variable, including those with fractional coefficients using an algebraic method

· LP3.04P - isolate a variable in formulas involving first degree terms (Worksheet included)

Assessment/Evaluation Techniques

· A variety of assessment tools should be used to properly evaluate the student.

· Through observation, make anecdotal comments on independent work, teamwork, organization skills, work habits, communication and initiative.

· Assess Part B with a written report rubric (see Appendix D).

· If the teacher wishes to have presentations instead of a written report, an verbal presentation rubric could be used (provided in Appendix C).

· Pencil and paper quiz would be used to assess if the student can set up a table, graph, find the intersection point and explain what it means in the context of another realistic situation.

Resources

Lappan, Glenda, et al. Moving Straight Ahead Linear Relationships. Dale Seymour Publication, 1988.

Working With Formulas (To be used in follow-up activity)

1. If you are borrowing money, you will have to pay some interest on the loan. The formula used to calculate interest is I = PRT/100. I is the amount of interest you must pay for borrowing “P” dollars at “R”% for “T” years.

a) Find the length of time needed to pay for a loan of $200 at an interest rate of 5% per year and an interest payment of $60

b) If you paid twice as much interest, on twice the amount of the loan at twice the interest rate, find the length of time you had the loan? What is your guess? Calculate the time. How close was your guess to the actual time?

c) If you paid three times as much interest, on three times the amount of the loan at the same interest rate as in part a), find the length of time you had the loan? Guess first! Then work out the actual time of the loan. How close was your guess to the actual time?

d) Rearrange this formula so that the time could be calculated with more ease or used on a spreadsheet.

e) Use the rearranged formula of part d) to find the time in the following situations

i) I = $50 R = 5% P = $500

ii) I = $50 R = 10% P = $500

2. Any temperature measured in Celsius degrees can be changed to Fahrenheit by using this formula: F=9C/5 + 32

Sue, a Canadian tourist visiting Florida has difficulty knowing what temperature it is since she is more familiar with Celsius scale. By rearranging the formula, she will have a quick method to convert Fahrenheit to Celsius degrees. Convert the following into Celsius degrees

i) 45° ii) 55° iii) 75°

3. If the volume of gas is heated so that its temperature increases by t degrees, then its volume increases, provided its pressure remains the same. There is a connection between the new volume V and the original volume v given by this formula: V = v(1 + t/273)

The change in temperature is in Celsius degrees. Rearrange the formula first to make repeated calculations easier. Find the temperature change in each situation

a) The original volume was 300 cm³ and the new volume is 600 cm³. Did the temperature increase or decrease?

b) The original volume was 600 cm³ and the new volume is 100 cm³. Did the temperature increase or decrease?

c) The original volume was 400 cm³ and the new volume is 950 cm³.

Activity 1.4: Bottled Water Dilemma

Time: 75 minutes

Description

This activity provides students with a context for exploring graphical representations of a linear system of equations in two variables and interpreting the point of intersection of two linear relations. Using graphs they will examine and compare two different reward structures, for supply and sale of bottled water to a school, offered by competing companies. They will be looking for information and results to help them make a recommendation regarding which company will better serve the school’s bottled water consumption and fund-raising needs. The introduction of a pricing war provides an interesting extension which allows students to see that, although both the equations and the graphs change, the solution remains the same. This provides a good lead into the algebraic solution of linear systems.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Linear Functions

Overall Expectations

LFV.02P - solve and interpret systems of two linear equations as they occur in applications.

Specific Expectations

LF2.01P - determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P - interpret the point of intersection of two linear relations within the context of a realistic situation.

Planning Notes

· The use of graphing tools, either graphing calculators or graphing software, is recommended since these allow students to apply and quickly see the effects of changes made to the reward schemes. They also enable students to easily and accurately find the coordinates of the point of intersection of the two lines representing the reward schedules.

· This activity would be more meaningful for the students if they were able to use real data. Perhaps your school is considering the installation of a vending machine. Your Student Council might be pleased to recruit the help of Grade 10 students in the decision making process.

Prior Knowledge Required

Number Sense and Algebra Grade 9

· add and subtract polynomials and multiply a polynomial by a monomial;

· expand and simplify polynomial expressions involving one variable.

Relationships Grade 9

· graph lines by hand or using graphing calculators or graphing software.

Teaching/Learning Strategies

Student Activity

Working in pairs, students will use graphing calculators or graphing software to investigate the two fund-raising schedules in the following problem.

Problem

The Student Council must decide between two companies tendering to supply bottled water in a vending machine for the cafeteria. They want to make sure they get the “best deal”. On offer are the following:

· Each month, Moose Country Water will pay the school five cents for every bottle sold after the first 1000 bottles.

· Northern Crystal Water will pay seven cents a bottle after 2000 have been sold each month.

Which company should the Student Council go with to raise as much money as possible?

1. Write a word equation and an algebraic equation to describe the relationship between funds raised and number of bottles sold under each company’s scheme.

2. Graph the equations and describe the graphs.

3. Use the graphs to compare the fund-raising possibilities under each scheme.

4. Based on the graphs, what advice would you give to the Student Council about which company they should choose? Explain the significance of the point of intersection of the graphs for the two companies?

5. Write your recommendations to Student Council in the form of a brief report. Include details of your graphical analysis.

Teacher Facilitation: If students are struggling with Questions 3 and 4, prompt them with probing questions like "Which company should they choose if, on average, 2500 bottles of water are bought from the machine each month? Which company should they choose if, on average, 5000 bottles are bought each month?

Extension: Have students consider the following development. Moose Country Water, in an effort to secure the contract, offers an additional incentive of a $50/month donation to the school fund. Upon hearing this, Northern Crystal immediately responds with a matching offer of a $50/month donation. How do these additional payments affect (a) the equations and graphs of funds raised vs. number of bottles? (b) your recommendations to Students Council?

Teacher Facilitation: Students may need more support with this part. They should notice that both lines shift up by the same amount, but that where they intersect, the number of bottles sold is the same as before the incentive donations were added.

Follow-up Skills: 150 minutes

· Finding the intersection of lines using graphing calculators or graphing software; solving contextual problems

Assessment/Evaluation Techniques

· Assessment in the Learning skills areas of independence and initiative is possible as students work on the activity.

· Individual student written reports will provide evidence of learning in all four Achievement Chart categories and could be evaluated using a rubric.

Accommodations

Pair students with reading or writing difficulties with students who may help them, by reading the instructions to them, and/or writing down their ideas and contributions for them. Make frequent checks on these students to ensure that they understand the concepts involved in the activity.

Resources

This activity is an adaptation of an example in “Algebra in a Technological World,” NCTM Addenda Series.

Activity 1.5: The Rewards of Design

Time: 150 minutes

Description

The analysis of wages for interior design work serves as a context for discussion of linear equations in the form Ax + By + C = 0. Students will then carry out a graphical investigation which will lead to solving equations by the elimination method.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands, and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Linear Functions

Overall Expectations

LFV.02P - solve and interpret systems of two linear equations as they occur in applications.

Specific Expectations

LF2.01P - determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P - interpret the point of intersection of two linear relations within the context of a realistic situation;

LF2.03P - solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;

LF2.04P - solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs;

LF3.02P - rearrange equations from the form y = mx + b to the form Ax + By + C = 0 and vice versa (if graphing calculators are used).

Planning Notes

· Graphing calculators or graphing software may be used for this activity. However, students need to be able to work with equations in the form Ax + By + C = 0 and so software such as Zap-a-Graph is preferred, especially for the exploration in the second part of the activity. Graph paper and pencil is also a good medium for this activity.

· Plan to have students work in pairs so that they may discuss what they see as they progress through the exploration.

Prior Knowledge Required

Analytic Geometry

· graph lines by hand (using a variety of techniques) and using technology

Number Sense and Algebra

· solving 1st-degree equations excluding equations with fractional coefficients, using an algebraic method

Teaching/Learning Strategies

Student Activity

As part of a career investigation into the field of interior design, Rima and Vladimir contacted the local bank and a nearby daycare, both of which had been recently renovated. The same interior design and paint team, consisting of a design consultant and an assistant who did the painting, had been used by both the bank and the daycare. The design team were paid $1380, by the bank, for a job which had required 6 hours of the consultant’s time and 30 hours of painting by the assistant. For the work they did at the daycare, the team was paid $920 for four hours of consultant’s time and 20 hours put in by the assistant.

Rima and Vladimir want to include the hourly rates of the consultant and the assistant in their report on interior design as a career.

1. Let x represent the hourly rate for the design consultant, and let y represent the hourly rate for the assistant. Complete the chart to obtain equations which will be in the form Ax + By + C = 0.

Bank				Daycare
	Consultant	Assistant	Total		Consultant	Assistant	Total
Hours				Hours
Hourly rate	x	y		Hourly rate	x	y
Earnings (hours × hourly rate)				Earnings

The equation for the bank renovation is _________________________

The equation for the daycare renovation is _________________________

2. Graph the above equations. What do the points on each line represent?

3. At what point do these lines intersect? What does this intersection point mean?

Teacher facilitation: When most students have completed this part of the activity, the teacher takes up the findings so far, as a whole class discussion. Emphasis should be placed on:

· the Ax + By + C = 0 form of the equation of a straight line;

· the interpretation of the point of intersection as the solution which gives the hourly rates for the consultant and the assistant.

Investigating Intersections

In this activity you will be discovering an algebraic process to find the intersection point. Let’s first look at some simple cases:

1. 7x - y = 2 2x - y = -3

(I) Graph the above equations.

(II) The coordinates of the point of intersection are:

(III) Add together the two equations above.

(IV) Graph the new equation on the same set of axes.

(V) What do you notice about these three lines?

2. 8x + 5y = 1 3x + 2y = 1

(I) Graph the above equations.

(II) The coordinates of the point of intersection are:

(III) Add together the two equations above.

(IV) Graph the new equation on the same set of axes.

(V) What do you notice about these three lines?

3. x - 4y = 6 -x - y = 4

(I) Graph the above equations.

(II) The coordinates of the point of intersection are:

(III) Add together the two equations above.

(IV) Graph the new equation on the same set of axes.

(V) What do you notice about these three lines?

4. In your own words, describe what is always true about the graph of the resulting equation when you add two equations together.

5. 2x + 3y = 6

(I) Graph the above equation.

(II) Multiply each term of the equation by 5. Graph the new equation on the same axes.

(III) Multiply each term of the equation by -4. Graph the new equation on the same axes.

(IV) Describe what you noticed about these graphs.

6. 2x + y = 3 À

x + y = 1 Á

(I) Graph these equations.

(II) Graph À + Á

(III) Graph À + 2 × Á

(IV) Graph À + (-2) × Á

(V) Describe what you notice about each of these lines.

(VI) Which lines, other than the original two, are the most important?

7. 3x - y = 2 Â

x + 2y = 10 Ã

(I) Graph the above equations.

(II) Graph  + Ã

(III) Graph 2 × Â + Ã

(IV) Graph  + (-3) × Ã

(V) Which lines, other than the original two are most important?

8. Without graphing, find the point of intersection of the following pairs of lines:

(I) 2x + 3y = 6 (II) x + 3y = -1

2x + y = -4 2x - y = 12

9. Recall the interior design problem you solved graphically:
The design team were paid $1380, by the bank, for a job which had required 6 hours of the consultant’s time and 30 hours of painting by the assistant. For the work they did at the daycare, the team was paid $920 for 4 hours of consultant’s time and 20 hours put in by the assistant.
Now use the algebraic method to find the hourly rates for the consultant and the assistant.

10. Revisit the bottled water problem:
The Student Council must decide between two companies tendering to supply bottled water in a vending machine for the cafeteria. On offer are the following:

· Each month, Moose Country Water will pay the school 5 cents for every bottle sold after the first 1000 bottles.

· Northern Crystal Water will pay 7 cents a bottle after 2000 have been sold each month.

Use an algebraic method to determine the point at which the amount of money raised would be the same for both companies.

Teacher Facilitation: Initially students will need help with the concept and process of adding two equations.

Follow-up Skills: 300 minutes

· Students will need to practise the algebraic method of solution by elimination. The teacher should provide realistic examples of systems for students to solve, as well as more problems in which it is necessary to formulate the system first.

· Following practice with elimination, students should solve systems by the algebraic method of substitution. Again, realistic applications should be provided.

· Practice with the algebra of changing from Ax + By = C form to y = mx + b form would also be appropriate following completion of these exercises.

Assessment/Evaluation Techniques

· A paper and pencil quiz which contains some questions of the type in Question 8 and a communication component in which students describe the algebraic method of solution they have discovered in the exploration.

Accommodations

· Pair students with reading or writing difficulties with other students who will be able to help them.

Resources

The above exploration is an adaptation of an activity in Activities for Active Learning and Teaching NCTM.

Activity 1.6: Battle of the Bands – Introduction to Quadratics

Time: 150 minutes

Description

In this activity, students will multiply linear expressions to obtain a quadratic function. Students will discover that quadratic functions are useful in solving optimization problems. They will use the regression menu from the graphing calculator to find the curve of best fit and find that the product of binomials is also an expression in the form of ax² + bx + c. The follow up to this activity will be expanding and simplifying polynomial expressions. The area high schools have decided to continue a band competition which was started last year. The High School Band Promoter has decided to change the ticket prices in order to maximize his revenue needed to pay for the advertising display in a mall close to the auditorium which he has rented for this event. He sets up spreadsheets to examine what is the best price to charge and what is the maximum floor space he can use at the mall.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands, and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a collaborative contributor who works effectively as an interdependent team member;

- a responsible citizen who accepts accountability for one’s own actions.

Strand(s): Quadratic Functions

Overall Expectations

QFV.03P - solve problems by interpreting graphs of quadratic functions.

Specific Expectations

QF3.02P - determine the maximum or minimum value of a quadratic functions from its graph, using graphing calculators or graphing software.

Planning Notes

· This activity should be done in pairs so students can discuss the problem and share ideas. However they should present an individual report with the aid of computer programs and/or graphing calculators.

· Remind students to ensure that window settings on graphing calculators are appropriate for activity.

Prior Knowledge Required

Relationships

· use of a graphing calculator or graphing program on a computer.

· construct tables of values and graph a non-linear relation derived from descriptions of realistic situations

Teaching/Learning Strategies

Teacher Facilitation: Discuss the need for a mathematical process for determining the price to charge and the amount of space to be used in a mall for an advertising display. There is a considerable amount of research and Mathematics which is used to determine the pricing and the amount of floor space to be allotted to any display. A guest speaker from a retail consultant firm would be helpful in talking about this to students and allowing them time to investigate possible jobs in this area. After some meaningful discussion, this activity can be presented to small groups of two or three students. Make the students aware that they will hand in individual reports, but are allowed to work together to discuss the problem and give support to each other.

Student Activity

Battle of the Bands – Student Handout

1. Doug, who is the promoter of this event has rented an auditorium to allow high school bands an opportunity to show their talent. Usually this event attracts 1000 people at $15 for each person. At this price, all the tickets were sold last year. Doug decided to set up a display at a local mall to encourage students to participate and register their bands. To pay for this display Doug was forced to increase the price of the ticket. In order to help him decide what would be a fair price to charge, he conducted a business survey and found that the number of tickets sold will decrease by 50 for every dollar increase.

a) Form a hypothesis about the best price to charge.

b) Complete the chart in your report.

Number of Increases	Ticket Price	Number of Tickets Sold	Revenue
0
1
2
3
4
. . .
x

c) State the revenue equation. Revenue = ( )( )

d) Use a graphing program and/or graphing calculator to create a graph of Revenue vs. Number of the Increases. Include the graph in your report. Don’t forget to set the window to get a better view of the relationship. Describe the shape of the graph in words.

e) What price should Doug charge for a ticket to maximize the revenue? Explain how you found the best price.

f) How many people would he expect to be in the auditorium with the new ticket price?

g) What is the maximum revenue? Explain how this can be found using the graph.

h) How much profit was made by the new price arrangement?

i) Enter the data from the chart in part b) onto the graphing calculator (L1 as Number of Increases and L2 as Revenue) to produce a scatter plot. Using the regression menu on your calculator find the equation of the curve of best fit through the points.

j) Compare the two equations from part c) and part i). Use your graphing calculator to input the two equations onto the Y= list. Do they represent the same curve? Explain why.

2. The local mall has given Doug an 8-metre rope for the perimeter of his display and will allow him to place the display in one of two places in the mall. In the north end of the mall, there is only one wall. He would have to use the rope for the other three sides. The south end of the mall has an area which consists of two walls and the rope would be used for two sides.

North end location South end location

a) Form a hypothesis about the best location and the best dimensions of the display area.

b) For each location set up a table with the following headings; Length of Display, Width of Display, Length of Rope, Area of Display. In your chart, include the general case (Let x be the length of the display). Include at least three diagrams of the display for each mall location.

c) State the area equation for each location. Area = ( )( )

d) For each location, construct a graph Area vs. Length of Display.

e) Which location gives the maximum area for the display. Explain how you decided which location was the best.

f) What is the maximum area? State the dimensions for the display.

g) Enter the data from your chart to produce a scatter plot on the calculator. Using the regression menu on your calculator, find the equation of the curve of best fit for both curves. How are these equations different from the ones you noted in part c)?

h) Input the two equations for the northern location (one from part c) and one from part g) on the calculator. Do they represent the same curve?

i) Input the two equations for the southern location on the calculator. Do they represent the same curve?

j) Why do two equations written in different form give the same curve?

Follow-up Skills: 150 minutes

Specific Expectations:

· QF1.01P - multiply two binomials and square a binomial

· QF1.02P - expand and simplify polynomial expressions involving the multiplying and squaring of binomials

Assessment/Evaluation Techniques

· Through observation, make anecdotal comments on independent work, teamwork, organizational skills, work habits, communication, and initiative.

· A Written Reports Rubric (Appendix D) can be used to evaluate such areas as clarity of communication and correctness of computation.

Accommodations

· Place students who are having difficulties with written work or language with students who will assist them. Extra time may be given for students demonstrating difficulties in language skills. Check on these students frequently to encourage them to stay on task.

Activity 1.7 Summative Assessment Activity

Time: 225 minutes

Description

This summative activity for Unit 1 will take place over three classes. Students will apply the various skills they have developed over the course of the unit to the practical problem of designing a deck and choosing an optimal method of construction.

Strand(s) and Expectations

Strand(s): Proportional Reasoning

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a self-directed, responsible, life-long learner who applies effective communication, decision-making, problem-solving, time and resource management skills;

- a collaborative contributor who works effectively as an interdependent team member; and who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others.

Overall Expectations

PRV.01P - solve problems derived from a variety of applications using proportional reasoning;

LFV.02P - solve and interpret systems of two linear equations as they occur in applications;

LFV.03P - manipulate algebraic expressions as they relate to linear functions.

Specific Expectations

PR1.01P - solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations);

PR1.02P - draw and interpret scale diagrams related to applications;

PR1.03P - distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning);

LF1.02P - construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software;

LF2.01P - determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P - interpret the point of intersection of two linear relations within the context of a realistic situation;

LF3.01P - write linear equations by generalizing from tables of values and by translating written descriptions.

Planning Notes

· Students should be placed in groups of three to design their decks and carry out the necessary calculations

· Deck designs may be completed using computer-aided design software or using graph paper

· Final written reports are to be completed individually

· Oral presentations will be completed by groups, but each student should play an active role in the presentation

· Students may consider using presentation software for their oral report

Prior Learning Required

· Students will have completed Unit 1.

Teaching/Learning Strategies

Student Activity

Your family is impressed with your practical knowledge in Mathematics and have assigned you the task of designing a deck for your home. They also have given you the task of choosing the most cost effective construction crew to build the deck.

Day 1: Designing the Deck

You are to design a one-level deck that will occupy no more than 15% of your backyard, which measures 50 feet by 65 feet.

(a) Construct a scale drawing of your deck.

You decide to submit the plans of your deck to two building crews. One crew is a local contracting company, “Decks R Us” and the other is a construction technology crew from your local high school. One crew requests that you submit the plans showing all measurements and area calculations in metric units; the other requests that all measurements and area calculations be submitted in Imperial units.

(b) Prepare the plans that will be submitted to each crew, using the requested criteria.

Day 2: Compare Estimates and Choose a Crew

Decks R Us will work at an hourly rate of $30 per hour, using a two-man crew. The five-man crew from the construction technology department at your high school will work for a $200 donation to the school and an additional $10 per hour. Materials are to be supplied by the owner.

(c) Using both algebraic and graphical models, provide a detailed costing comparison for each crew.

(d) Using the information provided by your teacher, determine the approximate cost of materials for the deck.

(e) Prepare a written report describing your recommendations for your family. Include your scale drawing, cost comparison, estimate of the time required to build your deck, a summary and rationale for your choice of construction crews, an estimate of the total material cost and a percentage breakdown of labour costs versus material costs for the job.

Day 3: Presenting your Report

Your family calls a meeting and asks you to present your findings in a five-minute seminar.

(f) Display your deck design; related graphs and costing analysis as you highlight the significant findings of your research into the deck design and construction process.

Assessment/Evaluation Techniques

· Written reports, verbal presentations, and group work can be assessed using the appropriate rubrics.

· Peer and self-evaluation can be completed using checklists and/or rubrics.

Assessment Activity: Pencil and Paper Test Sample Questions

1.

1 square = 3' x 3'

(a) Determine the area of the kitchen in square feet and square metres.

(b) Increase the dimensions of the kitchen by 15%.

(c) On average, it is estimated that home construction costs $95/square foot. What cost increase results when the dimensions of the kitchen are increased by 15%?

2. Stevie is planning a summer business doing minor repairs and carpentry work. He is trying to decide which payment schedule he should use to maximize his profits. He could charge a flat hourly rate of $15/hour, or he could charge a base fee of $25 plus an additional $10/hour.

(a) Graph the two scenarios on the same set of axes.

(b) Under what conditions is Plan A better than Plan B, and vice versa.

(c) Develop equations to represent each plan.

(d) What does the point of intersection represent?

(e) What would happen to the graph and to the point of intersection if he changed his base fee to $35?

(f) How much would he make under both plans if he worked for 8 hours?

(g) Explain which payment plan you would choose and why?

3. Develop a realistic problem that would describe a direct variation. Include the appropriate equation to describe the situation. Develop three relevant mathematical questions you could ask a classmate about your problem. (Provide answers to your questions.)

4. Sue was driving from London to Toronto as shown in the graph below:

(a) Determine an equation to represent the relationship in y = mx + b form for the first section of the graph.

(b) Write the equation in the form Ax + By + C = 0

(c) What does the slope of this graph represent?

(d) Sue stopped for lunch for ½ an hour. Continue the graph to represent this situation.

(e) Sue increased her speed by 15% for the last part of her trip. Draw this section of the graph to reflect her change in speed. Write a new equation to represent this change.

Continue to Unit 2 | Back to Course Overview | Back to Course Profiles main menu